Fine-grained quantum computational supremacy

Abstract

Output probability distributions of several sub-universal quantum computing models cannot be classically efficiently sampled unless some unlikely consequences occur in classical complexity theory, such as the collapse of the polynomial-time hierarchy. These results, so called quantum supremacy, however, do not rule out possibilities of super-polynomial-time classical simulations. In this paper, we study "fine-grained" version of quantum supremacy that excludes some exponential-time classical simulations. First, we focus on two sub-universal models, namely, the one-clean-qubit model (or the DQC1 model) and the HC1Q model. Assuming certain conjectures in fine-grained complexity theory, we show that for any a > 0 output probability distributions of these models cannot be classically sampled within a constant multiplicative error and in 2 (1−a)N+o(N) time, where N is the number of qubits. Next, we consider universal quantum computing. For example, we consider quantum computing over Clifford and T gates, and show that under another fine-grained complexity conjecture, output probability distributions of Clifford-T quantum computing cannot be classically sampled in 2 o(t) time within a constant multiplicative error, where t is the number of T gates.

Explicit lower bounds on strong simulation of quantum circuits in terms of $T$-gate count

“…With slight abuse of notation, we also allow the inverse of the T -gate (T † = P † T ) in the gate set and define the T -count of the circuit to be the number of T and T † gates altogether 2. We note that results similar to Theorem 1 have been obtained in[MT19] independently.…”